Essentially classical algorithms: Numerical methods to integrate the Schroedinger equation forward in time, as a particular PDE (tailored to account for the unitary character of evolution but sometime not even that): split-step methods, Krylov sub-spaces, Magnus schemes etc. Old fashioned but reliable tools like an old handy hammer is reliable when you need to strike a nail.
Tensor based classical algorithms. Initiated by the rise of many-body quantum physics in the late 1990 and rooted in the low-rank tensor approximation ideas, they marked a substantial advance in the fight with the Course of Dimensionality. Turned to be very successful when used to simulate many-body systems of chain-like topology characterized by a short-range entanglement; were a decisive factor in the development of many-body localization and transport in quantum disordered systems. But their time is over because most interesting quantum life is going on outside the bulb of weakly & shortly entangled states.
Digital quantum simulators. Qubitization and Trotterization are not new ideas (Lloyd? 1996?) but they were catalyzed by the QC hype; these two plus gates & measurements brought several digital Q-simulation algorithms. Interesting and inspiring though a bit formal and ‘fundamental’ (at the moment). Research activity and expectations on these new algorithms was going up steadily during last 6-7 years, being heated by the fast progress in the digital QC-technology and strong competition between the main players on the market (Rigetti, IBM, Google, and Microsoft). But if we see no new quantitative results and interesting model-specific simulations in the next few years – the algorithms will slide down from the Peak of Inflated Expectations (error correction and mitigation techniques are different story). A very natural field to to play with QC emulators.
Machine Learning & ANN algorithms. Train good your ANN and it will simulate your quantum model better than any algorithm from the two first generations! Good combination of hard-core quantum physics, AI/ML, provoking and partially controversial (previous post), these algorithms will certainly climb up the curve during next five-six years.